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Liouville theorems for an integral equation of Choquard type

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  • We establish sharp Liouville theorems for the integral equation

    $ u(x) = \int_{\mathbb{R}^n} \frac{u^{p-1}(y)}{|x-y|^{n-\alpha}} \int_{\mathbb{R}^n} \frac{u^p(z)}{|y-z|^{n-\beta}} dz dy, \quad x\in\mathbb{R}^n, $

    where $ 0<\alpha, \beta<n $ and $ p>1 $. Our results hold true for positive solutions under appropriate assumptions on $ p $ and integrability of the solutions. As a consequence, we derive a Liouville theorem for positive $ H^{\frac{\alpha}{2}}(\mathbb{R}^n) $ solutions of the higher fractional order Choquard type equation

    $ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * u^p\right) u^{p-1} \quad\text{ in } \mathbb{R}^n. $

    Mathematics Subject Classification: Primary: 35R11, 35J91; Secondary: 45G10, 35B53.


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  • [1] D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. 
    [2] P. d'Avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447–1476. doi: 10.1142/S0218202515500384.
    [3] P. BelchiorH. BuenoO. H. Miyagaki and G. A. Pereira, Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Anal., 164 (2017), 38-53.  doi: 10.1016/j.na.2017.08.005.
    [4] J. Bertoin, Lévy Processes, Cambridge University Press, 1996.
    [5] J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.
    [6] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.
    [7] D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994.  doi: 10.1017/prm.2018.67.
    [8] G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.
    [9] W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.
    [10] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.
    [11] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci., 29B (2009), 949-960.  doi: 10.1016/S0252-9602(09)60079-5.
    [12] P. Constantin, Euler Equations, Navier-Stokes Equations and Turbulence, Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of lecture Notes in Math. 1–43, Springer, Berlin, 2006. doi: 10.1007/11545989_1.
    [13] W. DaiY. FangJ. HuangY. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117.
    [14] P. Le, Symmetry and classification of solutions to an integral equation of Choquard type, submitted for publication.
    [15] P. Le, Liouville theorem and classification of positive solutions for a fractional Choquard type equation, Nonlinear Anal., 185 (2019), 123-141. doi: 10.1016/j.na.2019.03.006.
    [16] Y. Lei, Liouville theorems and classification results for a nonlocal Schrödinger equation, Discrete Contin. Dyn. Syst., 38 (2018), 5351-5377.  doi: 10.3934/dcds.2018236.
    [17] Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.  doi: 10.1007/s00209-012-1036-6.
    [18] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.
    [19] E. Lieb, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977) 185–194.
    [20] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.  doi: 10.1016/j.na.2009.01.014.
    [21] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.
    [22] I. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.
    [23] V. Moroz and J. V. Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.
    [24] V. Moroz and J. V. Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains}, J. Differential Equations, 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.
    [25] S. Pekar, Untersuchungen über die Elekronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.
    [26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970.
    [27] V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889.  doi: 10.1016/j.cnsns.2006.03.005.
    [28] D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.  doi: 10.1016/j.aml.2014.12.007.
    [29] W. Zhang and X. Wu, Nodal solutions for a fractional Choquard equation, J. Math. Anal. Appl., 464 (2018), 1167-1183.  doi: 10.1016/j.jmaa.2018.04.048.
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